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Compute the moment of inertia about the z-axis of the following objects with density, delta.

(a). The sphere $x^2+y^2+z^2\le a^2$

(b). The cylinder $x^2+y^2\le a^2$, $0\le z\le h$.

My attempt:

(a). Mass= $\frac{4}{3}\pi r^3$($\delta$)

radius of disk: $y=(a^2-x^2)^\frac{1}{2}$

volume of disk: $\pi(a^2-x^2)dx$

mass=(Density)(volume)=$(\delta)\pi(a^2-x^2)dx$ moment of inertia=[(mass of disk)(radius)^2]/2

[integral]: $[-a,a]$ $I=[\dfrac{(\pi)(\delta)}{2}][(a^2-x^2)^2dx]$

(kinda stuck here... not sure if this is right or in the right direction)

TEXTBOOK ANSWER: $\dfrac{2}{5}a^2(\delta\dfrac{4}{3}(\pi)a^3)=(\dfrac{2}{5}a^2)$(mass of sphere)

(b). radius=$a$

height=$h$, $0\le z\le h$ [integral]: $[0,M]$ I=[(dM)(a^2)/2] =[(a^2/2)(M)] =((a^2)/2)((pi)(a^2)(h)(delta))

\begin{equation}\begin{aligned} I&=[(dM)(a^2)]/2 \\ &= \dfrac{a^2}{2}M \\ &= (\dfrac{a^2}{2})(\pi a^2h)(\delta) \end{aligned}\end{equation} (again... I'm not confident that this is correct)

TEXTBOOK ANSWER: $\dfrac{a^2}{2}((\delta)(\pi)(a^2)(h))=(1/2)(a^2)$(mass of cylinder)

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  • $\begingroup$ It's generally expected that you will format your maths with MathJax. it takes a little time to learn but is really useful once you get the hang of it: math.meta.stackexchange.com/questions/5020/… (and it makes life a lot easier for any users wanting to help you, too). $\endgroup$ – user334732 Apr 25 '18 at 13:38
  • $\begingroup$ okay, thank you! $\endgroup$ – Emily J Bapst Apr 25 '18 at 16:32

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